How atoms of polycrystalline Nb20.6Mo21.7Ta15.6W21.1V21.0 refractory high-entropy alloys rearrange during the melting process

The melting mechanism of single crystal and polycrystalline Nb20.6Mo21.7Ta15.6W21.1V21.0 refractory high entropy alloys (RHEAs) were investigated by the molecular dynamics (MD) simulation using the second-nearest neighbor modified embedded-atom method (2NN MEAM) potential. For the single crystal RHEA, the density profile displays an abrupt drop from 11.25 to 11.00 g/cm3 at temperatures from 2910 to 2940 K, indicating all atoms begin significant local structural rearrangement. For polycrystalline RHEAs, a two-stage melting process is found. In the first melting stage, the melting of the grain boundary (GB) regions firstly occurs at the pre-melting temperature, which is relatively lower than the corresponding system-melting point. At the pre-melting temperature, most GB atoms have enough kinetic energies to leave their equilibrium positions, and then gradually induce the rearrangement of grain atoms close to GB. In the second melting stage at the melting point, most grain atoms have enough kinetic energies to rearrange, resulting in the chemical short-ranged order changes of all pairs.


Results and discussion
shows the variations of GB atom fractions and atomic binding energies of GB, grain, and system for Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEAs with the average grain size from 5.2 to 25.3 nm. It can be seen from Fig. 2 that the GB atom fraction decreases parabolically from 26.8 to 6.2% when the average grain size increases from 5.2 to 25.3 nm. In Fig. 1, according to the CNA results, atoms within the GB and grain are arranged in the undefined type and the BCC type, respectively. When the average grain size becomes smaller, the surface area to volume ratio of grains becomes higher. This result can also be seen for a nanoparticle, which the surface area to volume ratio becomes higher for a smaller nanoparticle. Accordingly, the fraction of GB atoms surrounding grains significantly increases for a smaller grain. Atoms at GB/grain interface possess higher local stresses and higher binding energy, so the atomic binding energy of GB, grain, and system decrease parabolically when the grain size increases from 5. The Warren-Cowley chemical short-range-order analysis 27 for Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEAs was used to quantify the attraction and repulsion between different elemental pairs and monitor the local structural rearrangement during the heating process. The chemical affinities of a referenced atom with its first neighbor atoms are evaluated by the Warren-Cowley short-range-order parameter, which can quantify the local short-ranged order. The definition of Warren-Cowley short-range-order parameter is shown in the following equation: where α ij is the short-ranged order parameter of the i-type referenced atom relative to j-type atom, N ij is the partial coordination number (CN) for the i-type referenced atom relative to j-type atom obtained from the (1)  www.nature.com/scientificreports/ predicted structure, and c j and N i are the fractions of j-type atom within the alloy and the average CN of i-type atoms, respectively. The value of c j by N i is an ideal partial CN for the referenced i-type atom relative to the first neighbor j-type atom, and this value completely depends on the respective atomic composition fraction of Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEA. The second term of Eq. (1) is the ratio of actual CN (N ij ) to ideal CN ( c j N i ) for the i-type reference atom to its first neighbor j-type atom. If this ratio is larger than 1, it means the affinity of j-type atom to i-type atom in the predicted structure is higher than that in the ideal structure. On the other hand, if this ratio is lower than 1, the affinity of j-type atom to i-type atom in the predicted structure is lower than that in the ideal structure. If the ratio is close to 1, it infers the affinity of j-type atom to i-type atom in the predicted structure is close to that in the ideal structure. Consequently, the positive and negative values of shortranged order indicate the lower and higher affinity of the element type pair, compared with their ideal affinity.
In previous related MD studies for HEA and BMG [28][29][30] , the Warren-Cowley parameters were used to quantify short-ranged order, indicating the affinity of the element type pair, compared with that of the corresponding element type pair in the ideal uniform distribution model. Figure 3 shows the short-ranged order distributions of all element type pairs for single crystal, grain sizes of 5.2 and 25.3 nm at 300 K. All atoms within these three structures are arranged according to the MaxEnt theory,  Profiles of GB fraction and the binding energies of grain, GB, and system for Nb 20 www.nature.com/scientificreports/ and the distance of the minimum between the first and second peaks of the radial distribution function (RDF) was used to determine short-ranged order values. In Fig. 3, short-ranged order values of pairs with the same element type are larger than 0.8, indicating all element types of Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEAs are arranged in the most uniform distribution. For pairs with different element types, all short-ranged order values are smaller than − 0.1, indicating the affinity of all pairs with different element types is much higher than those with the same element types. The first neighbor distance of a reference atom is required to calculate the Warren-Cowley short-ranged order parameter during the MD heating process, so the RDFs at different temperatures were calculated first for obtaining the distances at the minimum between the first and second peaks of RDFs. These distances are critical values to obtain the first neighbor atoms of a reference atom at different temperatures. Then the Warren-Cowley short-ranged order parameters of all element pairs at different temperatures were determined for reflecting the local short-ranged order changes. Figure 4a-c show the variations of density and enthalpy for single crystal, 25.3 nm, and 5.2 nm during the heating process. Because the enthalpies (densities) increase (decrease) linearly with the increasing temperature from 300 K to specific temperatures for all cases, the lowest temperatures of the horizontal axes in Fig. 4b,c start from higher values for clearly showing the variations of density and enthalpy near the melting points. For the single crystal, the enthalpy increases linearly with the increasing temperature from 300 to 2910 K and then shows an abrupt increase from 2910 to 2940 K, within which the local structure of single crystal Nb 20 31 . From 2940 to 3110 K, the enthalpy decreases parabolically, and then increases linearly with the increasing temperature when the temperature continuously increases from 3110 K. In Fig. 3 for the single crystal, the short-ranged order values of pairs with the same element type indicate the lowest affinity for atoms to the same element types before the melting point. When the system temperature is higher than the melting point of 2940 K, the atoms have enough kinetic energies to leave their equilibrium positions, and the elements possessing higher binding energies begin to aggregate together,  www.nature.com/scientificreports/ resulting in the decrease of enthalpy from 2940 to 3110 K. For the density profile, it decreases linearly with the increasing temperature from 300 to 2910 K, and then, from 2910 to 2940 K, the density displays an abrupt drop from 11.25 to 11.00 g/cm 3 , indicating the system undergoes the significant local structural rearrangement. In Fig. 4b,c, enthalpy profiles of 25.3 nm and 5.2 nm increase linearly with the increasing temperature from 2100 to 2900 K and from 1500 to 2540 K, respectively. When temperatures continuously increase from 2900 to 3100 K for 25.3 nm and from 2540 to 2700 K for 5.2 nm, enthalpies are almost unchanged. Within these temperature ranges, local structures undergo significantly rearrange. For the density profiles, the discontinuities at 2500 and 3100 K for 25.3 nm and the discontinuities at 2120 and 2860 K for 5.2 nm infer the local structural rearrangement smoothly proceeds during a wider temperature ranges, compared with the density drop within a narrow temperature range (2910-2940 K) for the single crystal. The temperatures of 2900 K and 2540 K are regarded as the melting points of 25.3 nm and 5.2 nm. Figure  where r i (0) is the position of the i-th atom at time 0, r i (t) represents the position of the ith atom at time t, and N is the total atom number in the system. The variation of SD is a sensitive parameter to investigate the extent of average atomic movement respect to a reference structure. Figure 6 illustrates the variations of SD values of all  Fig. 3 for the single crystal, one can see the short-ranged order values of pairs with the same element type are positive and larger than 0.95, while the short-ranged order values of pairs with different element types are negative. It is very complicated to investigate the short-ranged order values of all pairs during the heating process, so the average values of short-ranged order square for pairs with the same element type and with different element types were used to monitor the change of chemical short-ranged order during the heating process. For the single crystal Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEA, the average value of short-ranged order square of the same element type shown in Fig. 7 displays linearly decrease with the increasing temperature from 300 to 2530 K, and then decreases parabolically with the rising temperature from 2530 to 2920 K. The extent of lattice distortion and the increase in thermal vibration amplitude become more significant when the system temperature of single crystal Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEA continuously increases. Consequently, the distance between the first and second RDF peaks becomes closer when the temperature continuously increases, which can be seen in the inserts of RDF profiles at temperatures from 310 to 1500 K. At 1500 K, the first and second RDF peaks have been merged into a single peak. When the temperature continuously increases from 1500 to 2530 K, the first RDF peak becomes wider. It also leads to the linear decrease in the average value of short-ranged order square of the same element type. From 2530 to 2920 K, the value of the first RDF minimum increases with the increasing temperature. Some atoms have higher opportunity to contact with their second and third neighbor atoms with the same element type, leading to the significant decrease in the average short-ranged order square of the same element type from 2530 to 2920 K. At 2920 K, the average short-ranged order square of the same element type reaches its minimum, indicating the chemical short-ranged order of the same element types undergoes a critical change from less affinity to no preference. When the temperature increases from the melting point of 2940 K, the average short-ranged order square value of the same element type significantly rises. For different element types, the average value of short-ranged order square is relatively smaller and remains a constant at temperatures below the melting point of 2940 K. At temperatures above the melting point, the average value of short-ranged order square of different element pairs also displays an abrupt increase, which indicates the short-ranged order of different element types also undergoes a critical change at temperatures above 2940 K. Figure 8a demonstrates the short-ranged order distributions of all element type pairs at four characteristic temperatures, 2530, 2920 (minimum of short-ranged order square), 2940 (melting point), and 3110 K. At 2530 K,  Fig. 3. For investigating the short-ranged order change between 300 and 3110 K, Fig. 8b shows the differences of all short-ranged order pairs between 300 and 3110 K. The values in Fig. 8b are the short-ranged order values at 3110 K subtract the corresponding shortranged order values at 300 K. Consequently, the positive value of short-ranged order difference stands for the chemical affinity of an element type pair becomes weaker, whereas the negative value indicates the affinity of an element type pair becomes stronger. In Fig. 8b, one can see the short-ranged order differences of the same element type pairs are negative, indicating the affinity of the same element undergoes significant change at temperatures higher than the melting point. Figure 9 shows the atom distributions of Nb, Mo, W, Ta, and V within the single crystal Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEA at 3110 K. One can see the aggregation of Nb, Mo, W, Ta, and V is very obvious. For different element type pairs as shown in Fig. 8b, most of their short-ranged order differences are positive, indicating these element pairs become less affinity after the structural rearrangement after the melting. Figure 10a,b show the displacement vectors of all atoms at the melting temperature of 2940 K and 3110 K, respectively. The structure at 300 K is used as the reference structure to determine the atomic displacement vectors using OVITO. At 2940 K, one can see the displacement vector lengths of most atoms are longer than those in thermal vibration (marked in blue), indicating these atoms are far away from their equilibrium positions. At 3110 K, the displacement vector lengths of all atoms become much longer and the local structure is under significant change.
The SD and binding energy profiles of atoms within the grain and GB during the temperature elevation are shown in Fig. 11a,b for the case with the grain size of 25.3 nm, respectively. For the profiles of binding energy, both the values of grain and GB increase linearly with the increasing temperature from 2100 to 2800 K and from 2100 to 2670 K. Then binding energies of grain and GB display parabolic increase from 2800 to 2920 K and from 2670 to 2820 K, respectively. At temperatures higher than 2800 K for grain and 2670 K for GB, SD values begin to rise dramatically, indicating the local structural rearrangement occurs at these temperatures. It can see the temperature for GB structural rearrangement is relatively lower than that of grain. Consequently, at 2820 K, most GB atoms have enough kinetic energies to leave their equilibrium positions, and then these GB atoms gradually induce the rearrangement of grain atoms close to GB. Consequently, the temperature of 2820 K can be regarded as the pre-melting temperature, at which the melting of GB has been completed. When the temperatures continuously increase from 2920 to 3100 K for grain and from 2820 to 3100 K for GB, the binding energies decrease with the increasing temperature, and it indicates atoms of the same element types have a higher opportunity to contact each other. It should be noted the temperature, 2920 K (very close to the melting point obtained from enthalpy profile), is located at the binding energy peak of grain atoms, and it indicates most grain atoms have www.nature.com/scientificreports/ enough kinetic energies to rearrange. At temperature higher than 3100 K, the binding energies of grain and GB also illustrate linear increase with the increasing temperature. The average short-ranged order square profiles of the case with the average grain size of 25.3 nm during the heating process were shown in Fig. 12. RDF profiles at different temperatures are also shown in the inserts. One can see the value of the first RDF minimum become larger when the temperature continuously increases, leading to the decrease in the average short-ranged order square value. The minimum of the average short-ranged order square is at 2780 K as indicated by (I), below which the average short-ranged order square value of different element pairs is almost constant. When the temperature continuously increases from 2780 K to the melting point of 2900 K, both short-ranged order square profiles display linear increase with the increasing temperature. When the temperature increases from the melting point, these two short-ranged order square profiles begin the significant increase with the increasing temperature. The short-ranged order distributions for the Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEA with the average grain size of 25.3 nm at 2780 (minimum of average short-ranged order square), 2820 (pre-melting temperature), 2900 (melting point), and 2920 (binding energy peak of grain atom) K are illustrated in Fig. 13. At the pre-melting temperature, 2820 K, the GB atoms and some grain atoms close to GB have undergone significant local structural arrangement, leading to the short-ranged order changes of these atoms. The short-ranged order value variations of different element pairs at temperatures higher than the pre-melting temperature are very similar to those of single crystal Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEA at temperatures higher than that with the minimum of average short-ranged order square as shown in Fig. 8a.
For investigating the melting processes of GB and grain of the Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEA with the average grain size of 25.3 nm, the atom displacement vectors of 2820 (the pre-melting temperature), 2920 (20 K higher than the melting point), and 3100 K shown in Fig. 14a-c were used. The atom positions of the structure at 300 K were used as the reference, and the vectors were colored according to the length of a vector. In Fig. 14a, most GB atoms have larger displacement vector sizes (marked in red), and some grain atoms close to the GB atoms are also affected by the GB atoms, which possess larger displacement vector sizes (marked in green) as compared to those at the cores of grains (marked in blue). At 2920 K as shown in Fig. 14b, more grain atoms have large displacement vectors and the melting occurs toward the cores of grains. At 3100 K, the displacement  The SD and binding energy profiles of grain and GB atoms during the temperature elevation process are shown in Fig. 15a,b for the case with the average grain size of 5.2 nm, respectively. Variations of SD and binding energies with the increasing temperature are very similar to those of the case with the grain size of 25.3 nm as shown in Fig. 11a,b. For the case with a smaller average grain size, the pre-melting temperature of 2460 K is lower than that of 25.3 nm about 2820 K.
The average short-ranged order square profiles of the case with the average grain size of 5.2 nm during the heating process were shown in Fig. 16. RDF profiles at different temperatures are also shown in the inserts. The value of the first RDF minimum also becomes larger when the temperature continuously increases, leading to the decrease in the average short-ranged order square value. The minimum of the average short-ranged order square is at 2340 K as indicated by (I), below which the average short-ranged order square value of different element pairs is almost constant. When the temperature continuously increases from 2340 K to the pre-melting temperature of 2460 K, both short-ranged order square profiles slightly increase with the increasing temperature. Within this temperature range, most GB atoms have enough kinetic energies to rearrange, while most grain atoms still conduct the thermal vibration at their equilibrium positions. From the pre-melting temperature of 2460 K to the melting temperature of 2540 K, GB atoms leaving their equilibrium positions further affect grain atoms close to GB atoms to leave from their equilibrium positions. Consequently, these two short-ranged order square profiles increase significantly with the increasing temperature, revealing the short-ranged order of this RHEA undergoes considerable change. The short-ranged order distributions for the Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEA with the average grain size of 5.2 nm at 2340 (minimum of short-ranged order square), 2460 (pre-melting temperature), and 2540 (melting point and the binding energy peak of grain atom) K are illustrated in Fig. 17. Short-ranged order variations of different element pairs during the melting process are very similar to those shown in Fig. 13 for the case of 25.3 nm.  Fig. 18a-c. The structure at 2000 K was used as the reference for calculating the atomic displacement vectors. The atom positions of the structure at 300 K were used as the reference, and the vectors were colored according to the length of a vector. In Fig. 18a, GB atoms and grain atoms close to GB have longer displacement vector sizes (marked in red and green), as compared with those at the cores of grains (marked in blue). At 2540 K as shown in Fig. 18b, more grain atoms have large displacement vector lengths and the melting occurs toward the cores of grains. At 2700 K, the displacement vectors in Fig. 18c indicates all atoms within the Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEA with the average grain size of 5.2 nm leave their equilibrium positions and the system is in the melting state.    When the temperature increases from 2940 K, the enthalpy first becomes lower until 3110 K and then displays linear increase with the increasing temperature from 3110 K. When the system temperature is higher than the melting point of 2940 K, atoms have enough kinetic energies to leave their equilibrium positions, and the elements possessing higher binding energies begin to aggregate, resulting in the decrease of enthalpy from 2940 to 3110 K. For the melting mechanism of polycrystalline Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEAs, a two-stage melting process is proposed. The first melting stage is the melting of GB, and then the second melting stage is the melting of grains. In the first melting stage, the temperature for GB structural rearrangement is relatively lower than that of grain, and this temperature is the pre-melting temperature, at which most GB atoms have enough kinetic www.nature.com/scientificreports/ energies to leave their equilibrium positions, and then these GB atoms gradually induce the rearrangement of grain atoms close to GB. Pre-melting temperatures of grains of 25.3 nm and 5.2 nm are 2820 K and 2460 K, inferring the pre-melting temperature significantly depends on the average grain size. In the second melting stage at the melting point, most grain atoms have enough kinetic energies to rearrange, resulting in the shortranged order changes of all pairs. The CNA analysis results clearly indicate the local structures of GB atoms are amorphous with the undefined type of CNA result, and the fraction of GB atoms decreases with the increasing grain size. The average binding energy of amorphous structure is higher than that of crystal one within the grains. Consequently, the binding energy decreases with the increasing the grain size, resulting in a higher melting point. According to the melting points obtained by the MD simulation, the melting point and the grain size of polycrystalline  For the short-ranged order difference between 300 K and the melting point, the cases of single crystal and polycrystalline Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEAs are very similar. Short-ranged order differences of the same element type pairs are negative, indicating the affinity between the same element types undergoes significant change at temperatures higher than the melting point, resulting in the aggregation of Nb, Mo, W, Ta, and V. For different element type pairs, most of their short-ranged order differences are positive, implying that these element type pairs become less affinity after the structural rearrangement at temperatures higher than the melting point.    Table 1 lists the parameters of all single  elements 32 , and Tables 2 and 3 show all cross-element and ternary-element parameters of 2NN MEAM potential, parametrized by the reference data prepared by the density functional theory (DFT) calculation. The CASTEP plane along    Figure 1 shows the polycrystalline  (1) t (2) Fig. 1a-c are colored according to the element type, grain and grain boundary atoms identified by the common neighbor analysis (CNA) 35 , and the grain identity number, respectively.  Table 4. The maximum entropy (MaxEnt) theory 36 implemented by Monte Carlo (MC) method was used to have each compositional element undergo the most uniform distribution within all Nb 20.6 Mo 21.7 Ta 15.6 W 21.1 V 21.0 RHEAs, resulting in the maximum configurational entropy state. The element type of each nearest neighbor atom of a reference atom is not the same as the element type of the reference atom. Because all atoms were arranged by the Table 3. The parameter sets of C min and C max for ternary elements in the LAMMPS format.  21.0 RHEAs were investigated by the MD temperature elevation process from 300 to 3600 K. The heating process was processed in the increasing temperature by 10 K increment, and each increment was accompanied by a relaxation process in 10 ps before the subsequent temperature increases. For maintaining the constant temperature under the free stress during the temperature elevation process, the TtN method was utilized 37 . This method combines the Parrinello-Rahman variable shape size ensemble with the Nosé-Hoover thermostat. For the heating simulation, the periodic boundary conditions (PBCs) were used in all dimensions. Large-scale atomic/molecular massively parallel simulator (LAMMPS) was utilized to perform all MD simulations, which was developed by Plimpton et al. 38 . The OVITO package 39 was used to do all visualization and post process of all simulation results. Phys. Rev. B 59(5), 3527-3533 (1999). 38. Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117(1), 1-19 (1995